On problems without polynomial kernels

Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, Danny Hermelin

Research output: Contribution to journalArticlepeer-review

393 Scopus citations


Kernelization is a strong and widely-applied technique in parameterized complexity. A kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter in the input. A kernel is polynomial if the size and parameter of the output are polynomially-bounded by the parameter of the input. In this paper we develop a framework which allows showing that a wide range of FPT problems do not have polynomial kernels. Our evidence relies on hypothesis made in the classical world (i.e. non-parametric complexity), and revolves around a new type of algorithm for classical decision problems, called a distillation algorithm, which is of independent interest. Using the notion of distillation algorithms, we develop a generic lower-bound engine that allows us to show that a variety of FPT problems, fulfilling certain criteria, cannot have polynomial kernels unless the polynomial hierarchy collapses. These problems include k-Path, k-Cycle, k-Exact Cycle, k-Short Cheap Tour, k-Graph Minor Order Test, k-Cutwidth, k-Search Number, k-Pathwidth, k-Treewidth, k-Branchwidth, and several optimization problems parameterized by treewidth and other structural parameters.

Original languageEnglish
Pages (from-to)423-434
Number of pages12
JournalJournal of Computer and System Sciences
Issue number8
StatePublished - 1 Jan 2009
Externally publishedYes


  • Composition algorithm
  • Distillation algorithm
  • Kernelization
  • Lower bounds
  • Parameterized complexity
  • Polynomial hierarchy
  • Polynomial kernels

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Networks and Communications
  • Computational Theory and Mathematics
  • Applied Mathematics


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